Rotating disks and time dilation

I was just reading about the experimental proof of time dilation using some atomic clocks on aeroplanes. Which got me thinking...

Consider a rotating disk. Say it is a few hundred metres radius (although its size makes no difference to this question, it makes it easier for me to visualise). Clamp down an atomic clock onto the disk platter located at a point on the circumferential edge and call it point A. Strap a willing observer to that clock. Do the same again but put a clock and observer near to, but not at, the centre of the disk and call it point B. Put another clock and observer next to (but not on) the rotating disk and call it point C. Spin the disk as fast as possible for some long time and then slow to a halt.

1. There is no speed difference between point A and B i.e. A and B never move relative to each other. Therefore there will be no time difference between the clocks at A and B.

2. Points A and B do move relative to point C, but with different speed profiles. So, there will be different time difference between clocks A and C and clocks A and B.

Statement 1. says that clock reading are the same at A and B. Statement 2. implies that the clock readings at A and B are different. One or both statements can not be true.

What am I missing? (other than a better understanding of special relativity!)

1. There is no speed difference between point A and B i.e. A and B never move relative to each other.

This is only true in a co-rotating frame of reference. This frame is non-inertial and gravitational time dilation occurs there. Gravitational time dilation depends on the position in the non-inertial frame, not the relative movement.

I've done some reading since your reply. So, gravitational time dilation occurs because of the acceleration due to the rotating disk (just like gravity causes). Both A and B are accelerating at different rates because of their different positions on the disk. Your answer says that the frame is non-inertial, which is to say that it has acceleration. However, the accelerations A and B are different as measured from the coordinate frame of the disk. So, you are saying that the fact they are different means that they are different non-inertial frames. So, can I summarise by saying that because A and B undergo different accelerations wrt the same coordinate frame (the disk), they have different time dilations? And if so, what is the relationship between accn and time dilation?

I've done some reading since your reply. So, gravitational time dilation occurs because of the acceleration due to the rotating disk (just like gravity causes). Both A and B are accelerating at different rates because of their different positions on the disk. Your answer says that the frame is non-inertial, which is to say that it has acceleration. However, the accelerations A and B are different as measured from the coordinate frame of the disk. So, you are saying that the fact they are different means that they are different non-inertial frames. So, can I summarise by saying that because A and B undergo different accelerations wrt the same coordinate frame (the disk), they have different time dilations? And if so, what is the relationship between accn and time dilation?

And if the reference frame is not the disk, then what is it?

Acceleration does not affect time rates.

You can calculate the difference in time rates in the obvious way, from the speeds relative to an inertial frame, using Special Relativity. The disk is not an inertial frame.

You can also calculate the different time rates from the point of view of an observer on the disk as being due to the apparent gravitational potential, of which the acceleration is the gradient.

Time dilation due to acceleration/gravity is a more complicated thing than the time dilation due to a velocity difference. Consider e.g. two clocks attached to opposite ends of an accelerating steel rod. The rod will contract (in the inertial frame where it was originally at rest) as its speed increases, and that means that the clock at the rear always has a higher velocity than the clock at the front. So the cause of this "gravitational" time dilation is really just a velocity difference that arises because the clocks have to move in a certain way when they're attached to a rigid object.

So, can I summarise by saying that because A and B undergo different accelerations wrt the same coordinate frame (the disk), they have different time dilations?

Not quite:

1) They don't undergo coordinate accelerations w.r.t. the disc, they are at rest in the disc's frame. They undergo different proper accelerations, which are absolute not w.r.t. to some frame.

2) It is not necessary for them to undergo different proper accelerations, to have different clock rates. In an accelerating rocket a clock at the front can feel the same proper acceleration as a clock at the back, but the front clock ticks faster, in the frame of the rocket.

You can derive it via the redshift of light beam going from back to front or A to B. I think (not sure though) it goes like this: You use the instantaneous inertial frame of the emitter at emission time, and calculate the speed of the receiver in that frame at receive time. Then use the relativistic Doppler-Shift formula. The resulting frequency ratio is the time dilation.

The most useful fact you can learn that would help you understand things like this is that what a clock measures is the proper time of the curve in Minkowski space that represents its motion. Proper time is defined as the integral of [tex]\sqrt{dt^2-dx^2}[/tex] along the curve (where t and x are the coordinates of an inertial frame). You should think of this fact as an axiom that's a part of the definition of the theory. Note that in a 2+1-dimensional spacetime diagram, the motion of C is a straight line parallel to the t axis (dx=0 along the curve) and the motions of A and B are spirals (dx≠0).

1. There is no speed difference between point A and B i.e. A and B never move relative to each other. Therefore there will be no time difference between the clocks at A and B.

Your reasoning is in error here, but I grant you it's a tempting error.

Clock A is located on the axis of rotation, clock B is co-rotating with the disk's circumference, so yes, their relative distance does not change. But special relativity does not assert that if there is no change of relative distance there can be no time difference. Special relativity asserts that point B travels a longer distance. More precisely, special relativity asserts that there is in fact a difference in distance travelled between A and B.

The relativity comes in as follows: you can map all of the motion in a coordinate system that is co-moving with A, and then you can compute the distance traveled by B in that coordinate system. Alternatively, you can map all of the motion in a coordinate system in which A has a uniform velocity. For instance, you can let that coordinate system move along the disk's axis of rotation. Then the motion of A will be mapped as a straight line, and the motion of B will be mapped as a helix (a corkscrew).
Irrespective of how you map the motion (provided you map in an inertial coordinate system), the difference in distance traveled comes out the same.

Now, in order for B to travel a longer distance, and yet not fly away from A, B must undergo acceleration; there's no other way. But the difference in time cannot be attributed to the acceleration. Instead, the difference in time is correlated with the difference in distance travelled.